Snake lemma

The Snake lemma derives an exact sequence connecting kernels and cokernels from a commutative diagram with exact rows. Its connecting homomorphism is the key step.

Commutative diagram: A diagram of objects and maps in which every pair of paths with the same endpoints gives the same composite map. The lemma starts with a commutative diagram whose rows are exact.
Abelian category: A category with zero objects, kernels, cokernels, and image–coimage factorizations satisfying the abelian axioms. The lemma holds in abelian categories, not only for groups or modules.
Five lemma: A theorem that gives conditions under which the middle map in a commutative diagram of exact sequences is an isomorphism. It uses exact-sequence reasoning closely related to the Snake lemma.
Alexander Grothendieck: A French mathematician whose work reshaped algebraic geometry and introduced influential categorical methods. His work helped establish the broad categorical setting used for homological algebra.
Kernel: The subobject of inputs sent to zero by a map. Kernels of the vertical maps appear at the start of the resulting exact sequence.
Group homomorphism: A function between groups that preserves the group operation. Groups provide a familiar setting for the diagram and its induced maps.
Long exact sequence: An exact sequence with an extended chain of objects and maps, often induced by a short exact sequence. The Snake lemma is a standard route to long exact sequences in homology.
Homological algebra: The study of algebraic structures through chain complexes, exact sequences, and derived constructions. The Snake lemma is a basic theorem in this field.
Cokernel: The quotient of a map’s codomain by its image. Cokernels of the vertical maps appear at the end of the resulting exact sequence.
Module homomorphism: A map between modules that preserves addition and scalar multiplication. Modules are a standard setting where the lemma’s kernels and cokernels are computed.

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Zig-zag lemmaRelated: The same lift-and-boundary pattern underlies the connecting map in both results.